The proposed research is intended to develop mathematical and statistical methods for analyzing DNA sequence data from individuals sampled from several closely related populations. One part of the proposed research will be to develop methods that allow inference of past and present patterns of migration and admixture. The methods will be applied to DNA sequences from individuals from several human populations and from extinct human relatives, including Neanderthals and Denisovans (a distant relative of Neanderthals). The proposed research is intended to provide improved ways of understanding the history of humans and their close relatives. One goal will be to characterize the admixture that occurred between humans, Neanderthals, Denisovans, and other extinct human relatives. A second goal will be to identify genes in the human genome that recently increased in frequency under the effect of natural selection since the separation from Neanderthals and Denisovans but before the divergence of modern human populations. Although these methods are motivated by the recently available genomes from Neanderthals and Denisovans, they will be applicable to other groups of closely related species as more genomic data becomes available. The second part of the proposed research will be to develop methods for using DNA sequence data from a single species to detect evidence of a recent expansion in geographic range either into an empty habitat or a habitat already containing a resident admixing species. The goal will be to determine whether a range expansion has taken place, where its geographic origin was, and to infer details of the expansion, including the effective number of founders at the leading edge. These methods will be applied to data from several human populations in order to understand the recent history of population movement.